We optimize the proof technique from [1]} and
improve the upper bound of the number of maximal \(\alpha \) -gapped repeats in a word of length \(n\)  from \({\alpha n}\)  to \(18 \alpha n\) .
Unlike [1]}, [3]}, we partition the maximal \(\alpha \) -gapped repeats differently.
We categorize a gapped repeat (palindrome) depending on whether their left arm contains a periodic prefix or not.
Both subsets are treated differently.
For the ones having an periodic prefix, we think about the number of runs covering this prefix.
The other category is analyzed by using the results of secPointAnalysis.
We begin with a formal definition of both subsets and analyze the former subset.
